Yang-Lee zeros and the critical behavior of the infinite-range two- and three-state Potts models
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Yang-Lee zeros and the critical behavior of the infinite-range two- and three-statePotts models
Zvonko Glumac ∗ Department of Physics, Josip Juraj Strossmayer University of Osijek,P.O.B. 125, Trg Ljudevita Gaja 6, 31 000 Osijek, Croatia
Katarina Uzelac † Institute of Physics, P.O.B. 304, Bijenička 46, HR-10000 Zagreb, Croatia (Dated: February 26, 2018)The phase diagram of the two- and three-state Potts model with infinite-range interactions, inthe external field is analyzed by studying the partition function zeros in the complex field plane.The tricritical point of the three-state model is observed as the approach of the zeros to the real axisat the nonzero field value. Different regimes, involving several first- and second-order transitions ofthe complicated phase diagram of the three state model are identified from the scaling propertiesof the zeros closest to the real axis. The critical exponents related to the tricritical point and theYang-Lee edge singularity are well reproduced. Calculations are extended to the negative fields,where the exact implicit expression for the transition line is derived.
PACS numbers: 05.50.+q, 64.60.Cn
I. INTRODUCTION
Since the famous work of Yang and Lee [1, 2], the zerosof the partition function in a complex activity plane wereextensively studied as a useful tool in the investigation ofvarious aspects of phase transitions, up to giving rise toa new type of criticality related to complex parameterssuch as the Yang-Lee edge singularity [3]. Although orig-inally introduced in context of Ising model for explainingsingularities related to the second-order phase transition,zeros appeare to be useful for deriving the related criticalexponents and even distinguishing the nature of a phasetransition for a variety of models (see e.g. [4], or [5] fora recent review).In the case of the Ising model, where all the zeros liealong the circle in the complex plane, the correspondencebetween the density of zeros and thermodynamic poten-tials is simple and straightforward. For the Potts model[6] with an arbitrary number of states, the layout of Yang-Lee zeros is much more complicated [7–11] while evenmore intensive investigations were conducted on Fisherzeros [12, 13] in this model [14] and the zeros in thecomplex- q plane [15–18]. As far as the Yang-Lee zerosare concerned, it was shown analytically in one dimen-sion [19], and by numerical calculations on small systemsin higher dimensions [8, 10], that, for general q , thesezeros do not lie on the unit circle of the complex activ-ity plane, and that multiple lines, or bifurcations mayoccur [9]. This is related to a more complicated phasediagram of this model. When the number of states ofthe Potts model is larger than 2, the presence of real,positive external field h does not necessarily destroy the ∗ zglumac@fizika.unios.hr † [email protected] phase transition, but produces a line of first-order phasetransitions ending with the tricritical point. The modelis not symmetric under exchange h ↔ − h , and in thepresence of a negative field, a different and less explored[20] transition takes place. All this leads to a more com-plicated phase diagram in the temperature-field plane,that involves both continuous and first-order phase tran-sitions, and the tricritical points. In the complex activityplane, this is reflected by the fact that zeros do not lieon the unit circle, while more than one line of zeros thatapproach the real axis may appear.The main purpose of this paper is to present, in thespecial case of the three-state Potts model with infinite-range interactions, an integrated view covering differentcritical regimes of the entire phase diagram in terms ofthe behavior of the partition function zeros in the com-plex activity plane. We also report the results for theYang-Lee edge singularity above the tricritical tempera-ture of this model.The infinite-range Potts model is equivalent to themean-field (MF) approximation of the standard Pottsmodel with short-range (SR) interactions [21, 22], whichdescribes the critical beahaviour of the SR model cor-rectly only in sufficiently high dimensions, above theupper critical dimension. Nevertheless, its phase dia-gram displays similar diversity to that of the SR model,while it may be approached analytically, using the saddle-point approximation. In our approach we apply twice theHubbard-Stratonovich transformation to obtain the ex-act two-parameter analytic expression for the free energyof the infinite-range model. The same approach permitsus to obtain implicit analytic expressions for the parti-tion function zeros and to calculate zeros close to the realaxis in the finite system. The finite-size scaling analysisof a few zeros that lie closest to the real axis then allowsus to identify different critical regimes, and calculate therelated critical exponents.The outline of the paper is following. In Sec. II weillustrate the approach for the simple q = 2 case, whichis also used to obtain some estimates about the precisionof numerical calculations of the partition function zeros.The same methodology is then applied to the q = 3 modelin Sec. III. We first derive an exact two-parameter ex-pression for the free-energy density to be solved by asaddle-point method, and we discuss its shapes in thetwo-parameter space, related to different regimes of thephase diagram. In Secs. III A and III B, we derive ana-lytical expressions for the scaling properties of the lowestzeros in different critical regimes, for the cases with h ≥ and h < respectively. In Sec. III B 1 we present the nu-merical results for the partition function zeros, and thenumerical derivation of scaling exponents. Section IVcontains the concluding remarks. II. q = 2 (ISING) MODEL The energy of the q -state MF Potts model with N particles in the external field H has the form E = − J N N − X i =1 N X j = i +1 δ ( s i , s j ) − H N X i =1 δ ( s i , , (1)where s i denotes the particle at site i , which can occupyone of the q Potts states, while any two particles interactwith the same two-particle interaction J /N , regardlessof their distance.The reduced energy may be written in terms of num-bers of particles belonging to each of the q states, whichfor q = 2 are denoted by N and N − Ek B T = KN (cid:20)(cid:18) N (cid:19) + (cid:18) N (cid:19)(cid:21) + h N , (2)where K ≡ J k B T , h ≡ Hk B T = h + ı h , (3)and, by conservation of the total number of particles, N = N + N . The partition function Z N = N X N =0 (cid:18) NN (cid:19) e − E/ ( k B T ) , (4)after applying the Hubbard-Stratonovich transformation e A = 1 √ π Z + ∞−∞ e − y + A y √ d y, (5)with A = √ K N (cid:20) N N + 12 (cid:18) hK − (cid:19) (cid:21) , (6)turns into an integral Z N ∼ Z e − N f ( S ) dS, (7) TABLE I. Exact values of anomalous dimensions y h and y g as a function of K . K < K = 2 K > y h / / y g / / f ( S ) = K S S + 2) − ln h e KS + h i . (8)It may be solved by a standard saddle-point approxi-mation, which we recall here briefly. The correspondingextremal condition may be expressed by S = tanh KS + h , (9)where S = (2 N /N ) − is the order parameter, and themodel has a second-order phase transition for h = 0 and K = 2 .The density of partition function zeros, g , which all arelocated on the imaginary axis of the complex field plane,is related [23–25] to the jump in the order parameter S , S = 2 πg, (10)where S is the solution of Eq. (9). Above the criticaltemperature ( K < ) the gap free of zeros opens aroundthe real axis. At its edge h ,edge , which depends on tem-perature as [4] h ,edge = ± " r K − − K r K − , (11)another second-order phase transition in this model takesplace, known as the Yang-Lee edge singularity.The exact expressions for the critical exponents areknown [4, 22] for both transitions. We shall be inter-ested in scaling properties of the field h and the densityof zeros g for which the exact values of their anomalousdimensions in different critical regimes are summarizedin Table I. Note that the anomalous dimensions y h and y g are not defined here in the standard way. Due to theinfinite range of interactions the space scale has no mean-ing, so that the size scaling is carried out by scaling thenumber of particles instead. To to obtain the standardanomalous dimensions for this model as obtained whenapplying the mean-field approximation to the short-rangeversion of this model, one should multiply [18] the aboveexponents by d c , the upper critical dimension of the cor-responding phase transition (which is equal to 4 for theIsing transition and 6 for the Yang-Lee edge singularity).The anomalous dimension of the field is denoted by y h , which may be obtained from the scaling with size ofthe distance (in the parameter space) of the imaginarycomponent of the field h from its edge value, h edge h − h edge ∼ N − y h . (12) -0.001 -0.0005 0 0.0005 0.001 h h Numerical error
FIG. 1. (Color online) Solutions of the equations Re Z N = 0 [blue (dark gray) lines] and Im Z N = 0 [orange (light gray)lines, including the h = 0 axis] for N = 1 000 000 particles, at K = 2 . For distant points errors can be observed as deviationsof intersections from the imaginary axis. The exponent y g = y h σ denotes here the anomalous di-mension of the density of zeros, the power-law decay ofwhich is described near criticality by the exponent σg − g edge ∼ ( h − h edge ) σ ∼ (cid:16) N − y h (cid:17) σ ∼ N − y g . (13)Let us now sketch briefly the numerical derivation ofthese scaling properties by using the partition functionzeros closest to the real axis. A. Partition function zeros - Numerical results
For finite N , the partition function (4), is the polyno-mial of order N in the complex variable z = e h . Theusual root-finding routines soon become inapplicable forfinding the roots of Z N , as N increases. Instead, we lookfor the numerical solutions of the two equations Re Z N = 0 , Im Z N = 0 , (14)in a complex ( h , h ) plane. A typical example of layoutof these zeros, given by the intersections of dark graylines and light gray lines h = 0 , is presented in Fig. 1.Consistent with the Yang-Lee theorem, these zeros lieon the imaginary axis. As can be observed in Fig. 1, theprecision of such a calculation deteriorates for the remotezeros, which start to deviate from the imaginary axis, [7],as a consequence of the accumulation of numerical errors,which increase with N and with decreasing K . However,we do not have to deal with this problem here, sinceonly a few zeros closest to the real axis suffice to detectdifferent regimes and to analyze the corresponding singu-larities. Our analysis will thus be focused on these zeros,and include a comparison to the exact results allowingthe estimation of related errors.We calculated the position of the two zeros closest tothe real axis for a sequence of different sizes, ranging from TABLE II. Numerical ( h , extr ), and analytical ( h , edge ) valuesfor the Yang-Lee edge, followed by the numerical convergenceexponent y and its exact value y h from Table I. The errorbars are estimated to be of order of the last cited digit. Thenumerical part of the errors is at least one order of magnitudesmaller than the overall errors (including also the finite-sizecorrections). K h , extr h , edge y y h − . − . / . . . . . .
664 2 / . . . . . . / .
478 0 . . . . .
67 2 / up to particles. The distance of the zero closestto the real axis is used as an estimate of the positionof the Yang-Lee edge at a given temperature, while itsconvergence to its value in the thermodynamic limit isexpected, according to scaling property (12), to give theanomalous dimension y h h ( N ) = h , extr + const N y + · · · . (15)The results of extrapolating data by a simple least-squares fit are presented in Table II.The errors of the extrapolated values in Table II (aswell as in all the following tables) have two origins. Firstis the pure numerical error, which is relatively small (e.g.,for y it ranges from − to − with increasing K ) dueto the high accuracy ( − digits) of the input data.The second type of errors is due to the finite-size effects,and it comes from neglecting the higher order correctionterms in (15) and (18). They are in all cases one or moreorders of magnitude larger than the numerical errors. Inthe tables, the extrapolated values are presented up tothe first digit that differs from the exact value. Resultspresented in the second column clearly indicate the open-ing of the gap for K < K = 2 . Good agreement alsoexists between the convergence exponent y and the ex-actly known anomalous dimension y h of the field (fourthand fifth columns).The second quantity of interest is the edge density ofzeros, g N , defined as g N = 1 N d nd l . (16)The density of zeros near the Yang-Lee edge can be nu-merically calculated from the distance between the twozeros closest to the real axis, h (0) and h (1) , d l = (cid:20)(cid:16) h (1)1 − h (0)1 (cid:17) + (cid:16) h (1)2 − h (0)2 (cid:17) (cid:21) / , (17)using a discrete version of Eq. (16) with d n = 1 . It wasextrapolated to the limit N → ∞ by a least-squares fit TABLE III. Numerical ( g extr ) and analytical [ g exact , from Eq.(10)] values for the edge density of zeros, followed by thenumerical exponent y g, extr and its exact value y g (from TableI). The error bars are estimated to be of order of the last citeddigit. The numerical part of the errors is at least one order ofmagnitude smaller than the overall errors (including also thefinite-size corrections). K g extr g exact y g, extr y g . . . − . / − .
29 1 / − .
30 1 / − .
28 1 / to the form given by Eq. (13), i.e., g N = g extr + const N y g, extr . (18)The results of these extrapolations are presented inTable III. The vanishingly small numerical values of theedge density of zeros indicate a second-order transitionfor K ≤ . At lower temperatures, the transition is offirst order, and the edge density of zeros has a finite value.The critical exponent of the Yang-Lee edge singularity isin excellent agreement with the exact value for the Isingphase transition, given by the scaling at K = 2 . Thescaling exponent for K ≤ , corresponding to the Yang-Lee edge singularity, indicates a different critical regime,but is obtained with less precision. III. q = 3 MODEL
For the three-state Potts model in the symmetry break-ing field, the reduced energy may be written as − Ek B T = KN (cid:20)(cid:18) N (cid:19) + (cid:18) N (cid:19) + (cid:18) N (cid:19)(cid:21) + h N , (19) N + N + N = N, (20)where N , N , N denote numbers of particles in the threerespective states. The external field is conjugated hereto the Potts state and (dis)favors it, depending on thesign of h .Compared to the Ising case, this model has a muchricher phase diagram (Fig. 2), the details of which will bediscussed later in this section. In addition to the numberof exact results for h ≥ and arbitrary value of q , [22],we pay special attention to the less investigated h < part of the diagram.For h = 0 and q > , the Potts model has a first-orderphase transition at inverse temperature K ( q ) = 2 q − q − q − (21) K -0.5-0.4-0.3-0.2-0.100.10.2 h IsingKMSlow temperatures K trKMS K K trIsing disordered phase ordered phase, cond. in "1"ordered phase, cond. in "2" or "3"ord. ph., cond. in "1, 2" or "3"( I ) ( II )( III )( IV ) FIG. 2. (Color online) Phase diagram: Ising transition( h < ), open circles; KMS transition ( h > ), thick fullline. At low temperatures, K > K (3) , Ising and KMS linesare merged and a transition exists only for h = 0 , broken line. For q = 3 , K (3) = 4 ln 2 = 2 . . . . .While in the presence of an external magnetic field,there is no transition for q = 2 , models with q > conceala more complicated critical behavior, [26–29], with twotypes of field-driven transitions: for h > and for h < .For h > , there is a line of first-order transitions in ( K, h ) plane h ( K ) = 12 q − q − (cid:16) K ( q ) − K (cid:17) , K tr < K ≤ K ( q ) , (22)which starts at ( K ( q ) , and ends at the tricritical point ( K tr , h tr ) with K tr ( q ) = 4 q − q , h tr ( q ) = ln( q − − q − q , (23)where the transition is of second order. For q = 3 , K tr (3) = 8 / . ˙6 , h tr (3) = 0 . . . . and the transition line is given by thick full line in Fig.2.For negative values of h , the line of first-order transitions,also starts at ( K ( q ) , and approaches asymptoticallyto the point ( K ( q − , −∞ ) (open circles in Fig. 2). Theexact functional dependence h = h ( K ) is not known.For the purpose of numerical calculations, it is usefulto present the partition function in a polynomial form Z N = N X n =0 a n ( K ) z n , z = e h , (24)with the coefficients a n given by a n ( K ) = e K ( N − / N !( N − n ) ! e ( K/N )[ n − Nn ] (25) · n X m =0 e ( K/N )[ m − n m ] m ! ( n − m ) ! . On the other hand, for the analytical approach, the par-tition function, written as Z N = e K ( N − / e − ( N K/ / h/K − / +1] · N X N =0 (cid:18) NN (cid:19) N − N X N =0 (cid:18) N − N N (cid:19) (26) · e A e A , with A = √ N K (cid:18) N N + N N − (cid:19) ,A = r N K (cid:20) N N + 23 (cid:18) hK − (cid:19)(cid:21) , allows a twofold application of the Hubbard-Stratonovichtransformation (5) which reduces it to a double integral Z N ∼ Z + ∞− ∞ Z + ∞− ∞ d x d S e − N f ( x,S ) , (27)with the exact free-energy density f ( x, S ) − f ( x, S ) = − K (cid:16) x (cid:17) − K S ( S +1)+ln h xK e KS + h i , (28)where S = (3 N /N − / is the order parameter for h ≥ , and x = ( N − N ) /N is the order parameter for h < . The integral (27) is then solved by a saddle-pointapproach. The locations of minima are the solutions of atwo-by-two system of equations for x and Sx = 2 tanh( xK/
2) 1 e KS + h cosh( xK/
2) + 2 , (29) S = e KS + h cosh( xK/ − e KS + h cosh( xK/
2) + 2 . (30)The MF approaches to the general q -state Potts model,such as the solution by Kihara et al. [21] usually neglectthe fluctuations among states orthogonal to the orderedone, which corresponds to taking x = 0 . We examinehere the free energy in the entire ( x, S ) plane. In Figs. 3- 6 we illustrate the shapes of f for several characteris-tic values of temperature and the field, corresponding todifferent phases. For clarity in the figures, the maximaof − f are displayed instead of the minima of f .At low temperatures, the three maxima of the sameheight are obtained only for h = 0 [Fig. 3(a)]. Theycorrespond to a triply degenerate ordered phase of thesystem for h = 0 . By increasing temperature the max-imum at the origin (Fig. 3 b), which corresponds tothe disordered state, appears and starts to rise. For K (3) = 4 ln 2 = 2 . . . . , all the four maxima reachthe same height (Fig. 4), indicating the coexistence be-tween disordered and ordered phases at the first-ordertransition point ( K, h ) = ( K (3) , . At still higher tem-peratures, K < K (3) , the maxima behave differentlyfor positive and negative values of h (Fig. 5). For pos-itive values of the field, two maxima appear [Fig. 5(b)],which represent the transition that we denote here as theKMS transition, while for negative values of the field,there are three maxima [Fig. 5(a)] representing the tran-sition denoted here as Ising transition.At the KMS transition, the distance between the twomaxima diminishes as K goes to its tricritical value, untilthey eventually merge at ( K KMStr , h
KMStr ) given by (23)[Fig. 6(b)].The transition denoted as Ising, exists for all K (3) >K > K Isingtr = K (2) = 2 . At K Isingtr and h Isingtr = −∞ ,all three maxima [similar to those from the Fig. 6(a)],merge and the system undergoes a second-order phasetransition of an Ising universality class.For a summary of the critical behavior of the presentmodel let us turn back to the phase diagram in Fig. 2.There are four distinct phases:(I) Disordered phase at high temperatures,(II) For h > , there is an ordered phase in the Pottsstate . Ground state of this phase is non degenerate.Transition between this phase and disordered phase is offirst order (the thick full line in Fig. 2), except at the endpoint (tricritical point), where the transition is of secondorder, and where it is possible to go continuously fromthe phase I to the phase II.(III) For h = 0 , there is a line of first-order transitionsbetween the phase II and the phase IV (broken line inFig. 2). Its ground state is triply degenerate.(IV) For h < , the number of particles in state , cou-pled to the field, is suppressed favoring the particles inthe Potts states or . Ground state of this phase istwice degenerate. The first-order transition line separat-ing the phase IV and the disordered phase is marked bythe open circles in Fig. 2. This line has no endpointand goes down to h = −∞ . The continuous transitionbetween phases IV and the disordered phase is not pos-sible, except at infinity.Let us discuss first the scaling properties at differenttransitions. A. h > , KMS transitions A set of solutions satisfying the extrema conditions(29) and (30) corresponds to x = 0 , while S is the solution After the approximation of Kihara et al. , as explained in Sec.III A. See Sec. III B.
FIG. 3. (Color online) The free-energy density − f ( x, S ) at (a) ( K, h ) = (3 . , and (b) ( K, h ) = (2 . , . -0.5-0.25 0 0.25 0.5 x -0.5 -0.25 0 0.25 0.5 S FIG. 4. (Color online) The free-energy density − f ( x, S ) atinverse temperature K = K (3) = 4 ln 2 and h = 0 . of the equation S = e KS + h − e KS + h + 2 . (31)The same equation is found in the approximation used byKihara et al. [21, 27] to the second order phase transitionin the MF Potts model, which neglects the fluctuationswithin the remaining ( q − states. For h and K , relatedthrough the Eq. (22) (the thick full line in Fig. 2) h = ln 2 − K , (32)the positions of the extrema S ± [Fig. 5(b)] are of theform S ± = 14 ± ∆ S, (33)where ∆ S is a solution of the equation K ∆ S = ln 1 + ∆ S − ∆ S . (34) As the temperature rises,
K < K (3) , the distance be-tween the two maxima S ± decreases and eventually van-ishes at the tricritical point (23) K KMStr = 83 , h
KMStr = ln 2 − (35)(endpoint of the thick full line in Fig. 2).In the first-order transition regime and for large valuesof N , the dominant contribution to the partition function(27) comes from the two minima of equal depth Z N ∼ e − N f ( S + ) + e − N f ( S − ) ∼ e − N f ( S + ) n e − N [ f ( S − ) − f ( S + )] o . (36)Within this approximation, the calculation of zeros in theabove expression, reduces to the following set of equa-tions for the free-energy densities at S ± Re h f ( S − ) − f ( S + ) i = 0 , (37) N · Im h f ( S − ) − f ( S + ) i = (2 n + 1) π, n = 0 , , · · · To the leading order in /N , the solutions of the aboveequations are of the form h ( n )1 = ln 2 − K − N K O (cid:18) N (cid:19) , (38) h ( n )2 = 1∆ S
34 2 n + 1 N π + O (cid:18) N (cid:19) . Remark that, to the leading order in /N , the real partof zeros does not depend on n . Numerical calculations(Figs. 7 and 8) suggest that this remains true to all ordersin /N , i.e. that, at KMS transition, all the zeros lie ona straight line parallel to the imaginary axis.Calculation of the density of zeros at the Yang-Lee edgeis performed in the same way as for the Ising ( q = 2) case. FIG. 5. (Color online) The free-energy density − f ( x, S ) at K = 2 . and two values of the field: (a) h Ising = − . · · · and (b) h KMS = 0 . · · · .FIG. 6. (Color online) The free-energy density − f ( x, S ) at inverse temperature K = K KMStr = 8 / . ˙6 and external field:(a) h = − . · · · and (b) h = h KMStr = 0 . · · · . Relation (16) defines the density of zeros, while its valueat the edge is given by the distance between the two zerosclosest to the real axis. By inserting expansions (38) into(17), one gets g N = 2 ∆ S π + O (cid:18) N (cid:19) . (39)On the entire first-order transition line, ∆ S > making g finite. By approaching the tricritical point, ∆ S → ,and vanishing of g indicates the second-order transitionat the tricritical point. B. h < , Ising transitions Negative values of the field bring the particles in thePotts state (as defined in Eq. (19) ) to an energeticallyhigher level than the particles in the states and . Con-sequently, the system prefers to have most of the particlesin states and . In the limit h → −∞ , transitions of particles into state are completely forbidden and themodel reduces to the pure two-state (Ising) model with-out an external field. Thus, the negative field acts as achemical potential: it regulates the number of particlesin states and .In the limit h → −∞ the extrema conditions (29, 30)and the free-energy density (28), reduce to x = e x K − e x K + 1 , S = − . (40) − f ( x, − /
2) = − K (cid:16) x (cid:17) +ln h xK i + const. (41)Equation (40) has x → solution at K = K Isingtr = K (2) = 2 , so that the position of Ising tricritical pointis (cid:16) K Isingtr , h
Isingtr (cid:17) = (2 , − ∞ ) . (42)In the range of inverse temperatures K (2) < K < K (3) (open circles in Fig. 2), the free-energy density has ashape similar to the one shown in Fig. 6(a), and the tran-sition is of first order. By approaching K → K Isingtr = 2 ,the first order character of the transition becomes weakerand, at K = 2 , all three extrema of the free energy merge,and the transition changes its character into a second-order one.By studying the behavior of the free-energy densityaround the two edges of the Ising line of transitions, it ispossible to calculate the shape of h = h ( K ) in the twolimits. Close to the point ( K, h ) = ( K (3) , h ( K ) ≃ ln (cid:16) e K − (cid:17) . (43)Close to the point ( K, h ) = ( K (2) , − ∞ ) . h ( K ) ≃ ln (cid:16) K − (cid:17) + K . (44)In reference [27], authors discuss the cases with h < and q ≥ proving the existence of the transition in therange K ∈ (cid:16) K ( q − , K ( q ) (cid:17) , h ∈ (cid:16) − ∞ , (cid:17) . (45)Our results show explicitly how the above statement isextended to the case q = 3 .
1. Numerical results
The numerical calculation of Yang-Lee zeros startsfrom Eq. (24) Z N = N X n =0 a n ( K ) z n = Re Z N + ı Im Z N = 0 . (46)Numerical solutions of Re Z N = 0 [blue (dark gray) lines]and Im Z N = 0 [orange (light gray) lines] with theirintersections giving the positions of Yang-Lee zeros areillustrated in Figs. 7 and 8 for two typical values of K .Two lines of zeros, similar to those in Fig. 7, were alsoobserved in a numerical study [9], of a three-dimensional q = 25 Potts model with nearest-neighbour interactionson small lattices.Following the same procedure as the one presented inSec. II, we analyzed the positions of the two zeros clos-est to the real axis for a sequence of large system sizesranging from N = 10 up to sites. The position ofthe edge of the line of zeros and the edge density of ze-ros were extrapolated to the limit N → ∞ by using thesimple least-squares fit to the form h , ,N = h , , extr + const N y , , g N = g extr + const N y g . (47)The results of these extrapolations in the three differentregimes, corresponding to ( K < K (3) , h > , ( K ≥ K (3) , h = 0) , and ( K < K (3) , h < , are presented in Tables IV, V, and VI. The K < K (3) regime containstwo lines of transitions and in addition the Yang-Lee edgesingularity for sufficiently high temperatures so that thegap around the real axis is open . At lower temperatures, K ≥ K (3) , there is only one, h = 0 , line of zeros.Table IV contains data describing the h > or KMSline of transitions (thick full line in Fig. 2).At K ≥ K KMStr , the imaginary part of the field, is equalto zero, which means that the gap is closed at low temper-atures. At higher temperatures,
K < K
KMStr , the zerosaccumulate around the point ( h , h > which meansthat the gap is open with the Yang-Lee edge singularityat its edge. Real part of the edge still satisfies Eq. (32).Density of zeros at K > K
KMStr has a finite value, andthe transition is of first order. This finite value can becompared to the analytical expression (39) with high ac-curacy. At higher temperatures, K ≤ K KMStr , the edgedensity of zeros vanishes and the transition is of secondorder.The set of convergence exponents y ,extr and y g,extr hasthe same values as the corresponding exponents of the q = 2 model (Table I), showing that it belongs to thesame universality class. Since within the approximationby Kihara et al. the handling of fluctuations in the modelwith q > is essentially reduced to the calculation offluctuations in the two-state model, this result is not sur-prising.One may clearly distinguish the tricritical exponents for K = K tr from the Yang-Lee edge exponents obtained forhigher temperatures ( K < K tr ). The convergence expo-nent y ,extr is very close to , which is anticipated by Eq.(38) for the first order transitions. At K ≤ K KMStr , thetransition is of second order, and we have no analyticalexpression for y , but it seems that the value remainsfor all temperatures.At low temperatures, K ≥ K (3) (see Table V), zerosof the partition function are the result of the competitionbetween the three (at K > K (3) , Fig. 3) or the four (at K = K (3) , Fig. 4) extrema of f . The transition is ofa strong first-order type and loci of zeros form a straightline h = 0 (Figs. 8 a and 8 b). The convergence ex-ponents are equal to , except for the exponent of theedge density of zeros at K (3) . A qualitatively differenttransition, discussed in Sec. III B appears for h < (Ta-ble VI, open circles in Fig. 2). The numerical precisionof the results in Table VI is much poorer than that inTable IV. The main reason for that could be attributedto the fact that the corresponding zeros are fewer andmore spaced so that even the closest two zeros are muchmore distant from the real axis, than they are for anotherline. For this reason, we do not proceed here with thecalculation of the density of zeros used for other cases.The numerical extrapolations h , , extr and h , , exact pre-sented in Table VI yield the exact values [given by (29) In present work, we analyzed it only around h > tricriticalpoint. TABLE IV. The results of extrapolation for h > : real and imaginary parts of the closest zeros, h , , and the edge densityof zeros g , followed by the corresponding convergence exponents. The error bars are estimated to be of order of the last citeddigit. The numerical part of the errors is at least one order of magnitude smaller than the overall errors (including also thefinite-size corrections). K h , extr h , exact y , extr y , guess K (3) = 2 . · · · − . . . . · · · . K KMStr = 2 . ˙6 0 . . · · · . . . . · · · . . . . · · · . . . . · · · . . . . · · · . K h , extr h , exact y , extr y , guess K (3) = 2 . · · · − .
001 12 . − .
01 1 K KMStr = 2 . ˙6 10 − .
749 3 / . . − .
668 2 / . . − .
663 2 / . . − .
658 2 / . . − .
655 2 / K g extr g exact y g, extr y g, guess K (3) = 2 . · · · . . · · · .
997 12 . . . · · · .
94 1 K KMStr = 2 . ˙6 10 − . / . − .
28 1 / . − .
29 1 / . − .
335 1 / . − .
29 1 / TABLE V. The results of extrapolation for K ≥ K (3) , h = 0 : the real and imaginary part of the closest zeros, h , and theedge density of zeros g , followed by the corresponding convergence exponents. The error bars are estimated to be of order ofthe last cited digit. The numerical part of errors is by at least one order of magnitude smaller than the overall errors (includingalso the finite-size corrections). K h , extr h , exact y , extr y , guess . − .
001 13 . − .
007 1 K (3) = 4 ln 2 = 2 . · · · − .
05 1
K h , extr h , exact y , extr y , guess . − . . − .
004 1 K (3) = 4 ln 2 = 2 . · · · − .
04 1
K g extr g exact y g, extr y g, guess . . − .
997 13 . . − .
992 1 K (3) = 4 ln 2 = 2 . · · · . − .
53 3 / FIG. 7. (Color online) q = 3 and N = 3 000 . The positions of the part of the Yang-Lee zeros closest to the real field axis aregiven as the intersections of the blue (dark gray) and orange (light gray) lines: (a) K = 2 . and (b) K = 2 . . The broken blackline is the KMS value of the real field predicted by relation (32).FIG. 8. (Color online) q = 3 and N = 10 000 . The positions of the part of Yang-Lee zeros closest to the real field axis are givenas the intersections of the blue (dark gray) and orange (light gray) lines: (a) K = K (3) = 2 . . . . and (b) K = 3 . .TABLE VI. The results of extrapolation for h < : the real and imaginary part of the closest zeros, h , , followed by thecorresponding convergence exponents. The error bars are estimated to be of order of the last cited digit. The numerical partof errors is by at least one order of magnitude smaller than the overall errors (including also the finite-size corrections). K h , extr h , exact y , extr . − . − . · · · . . − . − . · · · . . − . − . · · · . K h , extr h , exact y , extr . − . . − . . − . y , , extr differ,however, significantly from the scaling exponent of thefirst-order phase transition, which should persist up to h = −∞ , as argued in Sec. III B. IV. CONCLUSION
The phase diagram of the two- and three-state Pottsmodel with infinite-range interactions in the external fieldwas analyzed by studying the partition function zeros inthe complex field plane.In the three-state case, we derived the exact two-pa-rameter expression for the free-energy density after atwofold application of the Hubbard-Stratonovich trans-formation. By applying the saddle-point approxima-tion in the two-parameter space, we reproduced well theknown analytical results in different regimes of the phasediagram with zero and positive external field. Calcula-tions could also be extended to the negative fields, whereanother, Ising-like, phase transition occurs. We derivedan implicit analytical expression and performed exact nu-merical calculation for the critical line of this Ising-liketransition, showing that it is of first order, and becomesof second order only in the limit h → −∞ , unlike the caseof the three-dimensional Potts model with short-range in-teractions, where the tricritical value of the field is finite[20].The same procedure was efficient for the study of thepartition function zeros in the complex field plane. Wehave shown, that in this model, with a rather complexphase diagram and with consequently more complicated loci of zeros including multiple lines, it is still possibleto identify different regimes of critical behavior from thesize scaling properties of the few zeros closest to the realaxis. We successfully identified the regime of the first-order phase transitions, and well reproduced the criticalexponents belonging to the two second-order phase tran-sitions, the tricritical point, and the Yang-Lee edge sin-gularity. In the case of the tricritical point, an unusualfeature is that the line of zeros reaches the real axis atthe point with nonzero field (i.e. loci of zeros lie on thenon unit circle in the complex activity plane). The Yang-Lee edge singularity could be observed for temperaturesabove the tricritical point temperature ( K < K tr ), andwas found to belong to the same universality class as thatof the MF Ising model.Due to the fact that the external field h breaks thesymmetry between Potts states of the q = 3 model inessentially the same way as for all q > models, one canexpect that the general shape of the phase diagram pre-sented in figure 2 as well as a major part of the obtainedresults could be extended to models with q > .It would also be interesting to perform a complemen-tary study of a more general case of the Yang-Lee zerosfor the Potts model with infinite range interactions withthe power-law decay, where some previous work has beenperformed [30] for the Ising model by the finite-rangescaling approach [31]. ACKNOWLEDGMENTS
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